LMFDB - Embedded newform 4005.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.9800860095$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.24698 q^{2} +3.04892 q^{4} -1.00000 q^{5} -1.55496 q^{7} +2.35690 q^{8} +O(q^{10})$	\(q+2.24698 q^{2} +3.04892 q^{4} -1.00000 q^{5} -1.55496 q^{7} +2.35690 q^{8} -2.24698 q^{10} +0.890084 q^{11} -0.198062 q^{13} -3.49396 q^{14} -0.801938 q^{16} -0.137063 q^{17} -7.60388 q^{19} -3.04892 q^{20} +2.00000 q^{22} -1.60388 q^{23} +1.00000 q^{25} -0.445042 q^{26} -4.74094 q^{28} +0.259061 q^{29} -8.09783 q^{31} -6.51573 q^{32} -0.307979 q^{34} +1.55496 q^{35} -0.137063 q^{37} -17.0858 q^{38} -2.35690 q^{40} +11.6528 q^{41} -5.24698 q^{43} +2.71379 q^{44} -3.60388 q^{46} -9.65279 q^{47} -4.58211 q^{49} +2.24698 q^{50} -0.603875 q^{52} +13.0368 q^{53} -0.890084 q^{55} -3.66487 q^{56} +0.582105 q^{58} -9.46011 q^{59} -5.10992 q^{61} -18.1957 q^{62} -13.0368 q^{64} +0.198062 q^{65} +5.38404 q^{67} -0.417895 q^{68} +3.49396 q^{70} -7.16421 q^{71} +0.317667 q^{73} -0.307979 q^{74} -23.1836 q^{76} -1.38404 q^{77} +12.4644 q^{79} +0.801938 q^{80} +26.1836 q^{82} -9.64742 q^{83} +0.137063 q^{85} -11.7899 q^{86} +2.09783 q^{88} +1.00000 q^{89} +0.307979 q^{91} -4.89008 q^{92} -21.6896 q^{94} +7.60388 q^{95} +0.439665 q^{97} -10.2959 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} - 3 q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 - 3 * q^5 - 5 * q^7 + 3 * q^8	$ 3 q + 2 q^{2} - 3 q^{5} - 5 q^{7} + 3 q^{8} - 2 q^{10} + 2 q^{11} - 5 q^{13} - q^{14} + 2 q^{16} + 5 q^{17} - 14 q^{19} + 6 q^{22} + 4 q^{23} + 3 q^{25} - q^{26} + 15 q^{29} - 6 q^{31} - 7 q^{32} - 6 q^{34} + 5 q^{35} + 5 q^{37} - 14 q^{38} - 3 q^{40} + 17 q^{41} - 11 q^{43} - 2 q^{46} - 11 q^{47} - 8 q^{49} + 2 q^{50} + 7 q^{52} + 11 q^{53} - 2 q^{55} - 12 q^{56} - 4 q^{58} - 3 q^{59} - 16 q^{61} - 18 q^{62} - 11 q^{64} + 5 q^{65} + 6 q^{67} - 7 q^{68} + q^{70} - 10 q^{71} - 16 q^{73} - 6 q^{74} - 14 q^{76} + 6 q^{77} - 7 q^{79} - 2 q^{80} + 23 q^{82} - 14 q^{83} - 5 q^{85} - 12 q^{86} - 12 q^{88} + 3 q^{89} + 6 q^{91} - 14 q^{92} - 19 q^{94} + 14 q^{95} + 4 q^{97} - 17 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 - 3 * q^5 - 5 * q^7 + 3 * q^8 - 2 * q^10 + 2 * q^11 - 5 * q^13 - q^14 + 2 * q^16 + 5 * q^17 - 14 * q^19 + 6 * q^22 + 4 * q^23 + 3 * q^25 - q^26 + 15 * q^29 - 6 * q^31 - 7 * q^32 - 6 * q^34 + 5 * q^35 + 5 * q^37 - 14 * q^38 - 3 * q^40 + 17 * q^41 - 11 * q^43 - 2 * q^46 - 11 * q^47 - 8 * q^49 + 2 * q^50 + 7 * q^52 + 11 * q^53 - 2 * q^55 - 12 * q^56 - 4 * q^58 - 3 * q^59 - 16 * q^61 - 18 * q^62 - 11 * q^64 + 5 * q^65 + 6 * q^67 - 7 * q^68 + q^70 - 10 * q^71 - 16 * q^73 - 6 * q^74 - 14 * q^76 + 6 * q^77 - 7 * q^79 - 2 * q^80 + 23 * q^82 - 14 * q^83 - 5 * q^85 - 12 * q^86 - 12 * q^88 + 3 * q^89 + 6 * q^91 - 14 * q^92 - 19 * q^94 + 14 * q^95 + 4 * q^97 - 17 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.24698	1.58885	0.794427	−	0.607359i	$-0.207771\pi$
$2$	2.24698	1.58885	0.794427	+	0.607359i	$0.207771\pi$
$3$	0	0
$3$	0	0
$4$	3.04892	1.52446
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.55496	−0.587719	−0.293859	−	0.955849i	$-0.594940\pi$
$7$	−1.55496	−0.587719	−0.293859	+	0.955849i	$0.594940\pi$
$8$	2.35690	0.833289
$9$	0	0
$10$	−2.24698	−0.710557
$11$	0.890084	0.268370	0.134185	−	0.990956i	$-0.457158\pi$
$11$	0.890084	0.268370	0.134185	+	0.990956i	$0.457158\pi$
$12$	0	0
$13$	−0.198062	−0.0549326	−0.0274663	−	0.999623i	$-0.508744\pi$
$13$	−0.198062	−0.0549326	−0.0274663	+	0.999623i	$0.508744\pi$
$14$	−3.49396	−0.933800
$15$	0	0
$16$	−0.801938	−0.200484
$17$	−0.137063	−0.0332427	−0.0166214	−	0.999862i	$-0.505291\pi$
$17$	−0.137063	−0.0332427	−0.0166214	+	0.999862i	$0.505291\pi$
$18$	0	0
$19$	−7.60388	−1.74445	−0.872224	−	0.489106i	$-0.837323\pi$
$19$	−7.60388	−1.74445	−0.872224	+	0.489106i	$0.837323\pi$
$20$	−3.04892	−0.681759
$21$	0	0
$22$	2.00000	0.426401
$23$	−1.60388	−0.334431	−0.167216	−	0.985920i	$-0.553478\pi$
$23$	−1.60388	−0.334431	−0.167216	+	0.985920i	$0.553478\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.445042	−0.0872799
$27$	0	0
$28$	−4.74094	−0.895953
$29$	0.259061	0.0481065	0.0240532	−	0.999711i	$-0.492343\pi$
$29$	0.259061	0.0481065	0.0240532	+	0.999711i	$0.492343\pi$
$30$	0	0
$31$	−8.09783	−1.45441	−0.727207	−	0.686418i	$-0.759182\pi$
$31$	−8.09783	−1.45441	−0.727207	+	0.686418i	$0.759182\pi$
$32$	−6.51573	−1.15183
$33$	0	0
$34$	−0.307979	−0.0528179
$35$	1.55496	0.262836
$36$	0	0
$37$	−0.137063	−0.0225331	−0.0112665	−	0.999937i	$-0.503586\pi$
$37$	−0.137063	−0.0225331	−0.0112665	+	0.999937i	$0.503586\pi$
$38$	−17.0858	−2.77168
$39$	0	0
$40$	−2.35690	−0.372658
$41$	11.6528	1.81986	0.909930	−	0.414761i	$-0.136135\pi$
$41$	11.6528	1.81986	0.909930	+	0.414761i	$0.136135\pi$
$42$	0	0
$43$	−5.24698	−0.800157	−0.400078	−	0.916481i	$-0.631017\pi$
$43$	−5.24698	−0.800157	−0.400078	+	0.916481i	$0.631017\pi$
$44$	2.71379	0.409119
$45$	0	0
$46$	−3.60388	−0.531362
$47$	−9.65279	−1.40800	−0.704002	−	0.710198i	$-0.748606\pi$
$47$	−9.65279	−1.40800	−0.704002	+	0.710198i	$0.748606\pi$
$48$	0	0
$49$	−4.58211	−0.654586
$50$	2.24698	0.317771
$51$	0	0
$52$	−0.603875	−0.0837425
$53$	13.0368	1.79075	0.895374	−	0.445316i	$-0.146909\pi$
$53$	13.0368	1.79075	0.895374	+	0.445316i	$0.146909\pi$
$54$	0	0
$55$	−0.890084	−0.120019
$56$	−3.66487	−0.489739
$57$	0	0
$58$	0.582105	0.0764342
$59$	−9.46011	−1.23160	−0.615801	−	0.787902i	$-0.711167\pi$
$59$	−9.46011	−1.23160	−0.615801	+	0.787902i	$0.711167\pi$
$60$	0	0
$61$	−5.10992	−0.654258	−0.327129	−	0.944980i	$-0.606081\pi$
$61$	−5.10992	−0.654258	−0.327129	+	0.944980i	$0.606081\pi$
$62$	−18.1957	−2.31085
$63$	0	0
$64$	−13.0368	−1.62960
$65$	0.198062	0.0245666
$66$	0	0
$67$	5.38404	0.657766	0.328883	−	0.944371i	$-0.393328\pi$
$67$	5.38404	0.657766	0.328883	+	0.944371i	$0.393328\pi$
$68$	−0.417895	−0.0506772
$69$	0	0
$70$	3.49396	0.417608
$71$	−7.16421	−0.850235	−0.425118	−	0.905138i	$-0.639767\pi$
$71$	−7.16421	−0.850235	−0.425118	+	0.905138i	$0.639767\pi$
$72$	0	0
$73$	0.317667	0.0371801	0.0185901	−	0.999827i	$-0.494082\pi$
$73$	0.317667	0.0371801	0.0185901	+	0.999827i	$0.494082\pi$
$74$	−0.307979	−0.0358018
$75$	0	0
$76$	−23.1836	−2.65934
$77$	−1.38404	−0.157726
$78$	0	0
$79$	12.4644	1.40236	0.701178	−	0.712986i	$-0.252658\pi$
$79$	12.4644	1.40236	0.701178	+	0.712986i	$0.252658\pi$
$80$	0.801938	0.0896594
$81$	0	0
$82$	26.1836	2.89149
$83$	−9.64742	−1.05894	−0.529471	−	0.848328i	$-0.677609\pi$
$83$	−9.64742	−1.05894	−0.529471	+	0.848328i	$0.677609\pi$
$84$	0	0
$85$	0.137063	0.0148666
$86$	−11.7899	−1.27133
$87$	0	0
$88$	2.09783	0.223630
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	0.307979	0.0322849
$92$	−4.89008	−0.509826
$93$	0	0
$94$	−21.6896	−2.23711
$95$	7.60388	0.780141
$96$	0	0
$97$	0.439665	0.0446412	0.0223206	−	0.999751i	$-0.492895\pi$
$97$	0.439665	0.0446412	0.0223206	+	0.999751i	$0.492895\pi$
$98$	−10.2959	−1.04004
$99$	0	0
$100$	3.04892	0.304892
$101$	15.7778	1.56995	0.784974	−	0.619529i	$-0.212676\pi$
$101$	15.7778	1.56995	0.784974	+	0.619529i	$0.212676\pi$
$102$	0	0
$103$	−16.4916	−1.62496	−0.812481	−	0.582987i	$-0.801884\pi$
$103$	−16.4916	−1.62496	−0.812481	+	0.582987i	$0.801884\pi$
$104$	−0.466812	−0.0457747
$105$	0	0
$106$	29.2935	2.84524
$107$	−5.96077	−0.576250	−0.288125	−	0.957593i	$-0.593032\pi$
$107$	−5.96077	−0.576250	−0.288125	+	0.957593i	$0.593032\pi$
$108$	0	0
$109$	−1.75840	−0.168424	−0.0842120	−	0.996448i	$-0.526837\pi$
$109$	−1.75840	−0.168424	−0.0842120	+	0.996448i	$0.526837\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	1.24698	0.117828
$113$	10.8116	1.01707	0.508536	−	0.861041i	$-0.330187\pi$
$113$	10.8116	1.01707	0.508536	+	0.861041i	$0.330187\pi$
$114$	0	0
$115$	1.60388	0.149562
$116$	0.789856	0.0733363
$117$	0	0
$118$	−21.2567	−1.95683
$119$	0.213128	0.0195374
$120$	0	0
$121$	−10.2078	−0.927977
$122$	−11.4819	−1.03952
$123$	0	0
$124$	−24.6896	−2.21719
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.8823	−1.58680	−0.793399	−	0.608701i	$-0.791691\pi$
$127$	−17.8823	−1.58680	−0.793399	+	0.608701i	$0.791691\pi$
$128$	−16.2620	−1.43738
$129$	0	0
$130$	0.445042	0.0390328
$131$	5.26205	0.459747	0.229874	−	0.973221i	$-0.426169\pi$
$131$	5.26205	0.459747	0.229874	+	0.973221i	$0.426169\pi$
$132$	0	0
$133$	11.8237	1.02525
$134$	12.0978	1.04509
$135$	0	0
$136$	−0.323044	−0.0277008
$137$	−16.7875	−1.43425	−0.717125	−	0.696945i	$-0.754542\pi$
$137$	−16.7875	−1.43425	−0.717125	+	0.696945i	$0.754542\pi$
$138$	0	0
$139$	6.88769	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$139$	6.88769	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$140$	4.74094	0.400682
$141$	0	0
$142$	−16.0978	−1.35090
$143$	−0.176292	−0.0147423
$144$	0	0
$145$	−0.259061	−0.0215139
$146$	0.713792	0.0590738
$147$	0	0
$148$	−0.417895	−0.0343507
$149$	19.9758	1.63648	0.818242	−	0.574874i	$-0.194949\pi$
$149$	19.9758	1.63648	0.818242	+	0.574874i	$0.194949\pi$
$150$	0	0
$151$	0.274127	0.0223081	0.0111541	−	0.999938i	$-0.496449\pi$
$151$	0.274127	0.0223081	0.0111541	+	0.999938i	$0.496449\pi$
$152$	−17.9215	−1.45363
$153$	0	0
$154$	−3.10992	−0.250604
$155$	8.09783	0.650434
$156$	0	0
$157$	−4.86592	−0.388343	−0.194171	−	0.980968i	$-0.562202\pi$
$157$	−4.86592	−0.388343	−0.194171	+	0.980968i	$0.562202\pi$
$158$	28.0073	2.22814
$159$	0	0
$160$	6.51573	0.515114
$161$	2.49396	0.196552
$162$	0	0
$163$	7.30798	0.572405	0.286203	−	0.958169i	$-0.407607\pi$
$163$	7.30798	0.572405	0.286203	+	0.958169i	$0.407607\pi$
$164$	35.5284	2.77430
$165$	0	0
$166$	−21.6775	−1.68250
$167$	11.5036	0.890179	0.445089	−	0.895486i	$-0.353172\pi$
$167$	11.5036	0.890179	0.445089	+	0.895486i	$0.353172\pi$
$168$	0	0
$169$	−12.9608	−0.996982
$170$	0.307979	0.0236209
$171$	0	0
$172$	−15.9976	−1.21981
$173$	2.61356	0.198706	0.0993528	−	0.995052i	$-0.468323\pi$
$173$	2.61356	0.198706	0.0993528	+	0.995052i	$0.468323\pi$
$174$	0	0
$175$	−1.55496	−0.117544
$176$	−0.713792	−0.0538041
$177$	0	0
$178$	2.24698	0.168418
$179$	9.74525	0.728394	0.364197	−	0.931322i	$-0.381343\pi$
$179$	9.74525	0.728394	0.364197	+	0.931322i	$0.381343\pi$
$180$	0	0
$181$	−9.54958	−0.709815	−0.354907	−	0.934901i	$-0.615488\pi$
$181$	−9.54958	−0.709815	−0.354907	+	0.934901i	$0.615488\pi$
$182$	0.692021	0.0512960
$183$	0	0
$184$	−3.78017	−0.278678
$185$	0.137063	0.0100771
$186$	0	0
$187$	−0.121998	−0.00892137
$188$	−29.4306	−2.14644
$189$	0	0
$190$	17.0858	1.23953
$191$	−7.52781	−0.544693	−0.272347	−	0.962199i	$-0.587800\pi$
$191$	−7.52781	−0.544693	−0.272347	+	0.962199i	$0.587800\pi$
$192$	0	0
$193$	5.03684	0.362559	0.181280	−	0.983432i	$-0.441976\pi$
$193$	5.03684	0.362559	0.181280	+	0.983432i	$0.441976\pi$
$194$	0.987918	0.0709284
$195$	0	0
$196$	−13.9705	−0.997890
$197$	0.263373	0.0187646	0.00938228	−	0.999956i	$-0.497013\pi$
$197$	0.263373	0.0187646	0.00938228	+	0.999956i	$0.497013\pi$
$198$	0	0
$199$	−15.8931	−1.12663	−0.563315	−	0.826242i	$-0.690474\pi$
$199$	−15.8931	−1.12663	−0.563315	+	0.826242i	$0.690474\pi$
$200$	2.35690	0.166658
$201$	0	0
$202$	35.4523	2.49442
$203$	−0.402829	−0.0282731
$204$	0	0
$205$	−11.6528	−0.813866
$206$	−37.0562	−2.58183
$207$	0	0
$208$	0.158834	0.0110131
$209$	−6.76809	−0.468158
$210$	0	0
$211$	−5.87800	−0.404658	−0.202329	−	0.979318i	$-0.564851\pi$
$211$	−5.87800	−0.404658	−0.202329	+	0.979318i	$0.564851\pi$
$212$	39.7482	2.72992
$213$	0	0
$214$	−13.3937	−0.915577
$215$	5.24698	0.357841
$216$	0	0
$217$	12.5918	0.854787
$218$	−3.95108	−0.267601
$219$	0	0
$220$	−2.71379	−0.182964
$221$	0.0271471	0.00182611
$222$	0	0
$223$	17.7995	1.19195	0.595973	−	0.803005i	$-0.296767\pi$
$223$	17.7995	1.19195	0.595973	+	0.803005i	$0.296767\pi$
$224$	10.1317	0.676952
$225$	0	0
$226$	24.2935	1.61598
$227$	−6.34721	−0.421279	−0.210639	−	0.977564i	$-0.567555\pi$
$227$	−6.34721	−0.421279	−0.210639	+	0.977564i	$0.567555\pi$
$228$	0	0
$229$	15.8974	1.05053	0.525264	−	0.850939i	$-0.323966\pi$
$229$	15.8974	1.05053	0.525264	+	0.850939i	$0.323966\pi$
$230$	3.60388	0.237633
$231$	0	0
$232$	0.610580	0.0400866
$233$	11.9976	0.785989	0.392995	−	0.919541i	$-0.371439\pi$
$233$	11.9976	0.785989	0.392995	+	0.919541i	$0.371439\pi$
$234$	0	0
$235$	9.65279	0.629679
$236$	−28.8431	−1.87752
$237$	0	0
$238$	0.478894	0.0310421
$239$	−5.59850	−0.362137	−0.181068	−	0.983471i	$-0.557956\pi$
$239$	−5.59850	−0.362137	−0.181068	+	0.983471i	$0.557956\pi$
$240$	0	0
$241$	−11.8780	−0.765129	−0.382565	−	0.923929i	$-0.624959\pi$
$241$	−11.8780	−0.765129	−0.382565	+	0.923929i	$0.624959\pi$
$242$	−22.9366	−1.47442
$243$	0	0
$244$	−15.5797	−0.997389
$245$	4.58211	0.292740
$246$	0	0
$247$	1.50604	0.0958271
$248$	−19.0858	−1.21195
$249$	0	0
$250$	−2.24698	−0.142111
$251$	−16.1957	−1.02226	−0.511131	−	0.859503i	$-0.670773\pi$
$251$	−16.1957	−1.02226	−0.511131	+	0.859503i	$0.670773\pi$
$252$	0	0
$253$	−1.42758	−0.0897514
$254$	−40.1812	−2.52119
$255$	0	0
$256$	−10.4668	−0.654176
$257$	−1.37867	−0.0859988	−0.0429994	−	0.999075i	$-0.513691\pi$
$257$	−1.37867	−0.0859988	−0.0429994	+	0.999075i	$0.513691\pi$
$258$	0	0
$259$	0.213128	0.0132431
$260$	0.603875	0.0374508
$261$	0	0
$262$	11.8237	0.730471
$263$	30.9788	1.91024	0.955118	−	0.296226i	$-0.0957281\pi$
$263$	30.9788	1.91024	0.955118	+	0.296226i	$0.0957281\pi$
$264$	0	0
$265$	−13.0368	−0.800846
$266$	26.5676	1.62897
$267$	0	0
$268$	16.4155	1.00274
$269$	−22.0301	−1.34320	−0.671600	−	0.740914i	$-0.734393\pi$
$269$	−22.0301	−1.34320	−0.671600	+	0.740914i	$0.734393\pi$
$270$	0	0
$271$	25.4523	1.54612	0.773060	−	0.634333i	$-0.218725\pi$
$271$	25.4523	1.54612	0.773060	+	0.634333i	$0.218725\pi$
$272$	0.109916	0.00666465
$273$	0	0
$274$	−37.7211	−2.27881
$275$	0.890084	0.0536741
$276$	0	0
$277$	12.1521	0.730151	0.365075	−	0.930978i	$-0.381043\pi$
$277$	12.1521	0.730151	0.365075	+	0.930978i	$0.381043\pi$
$278$	15.4765	0.928219
$279$	0	0
$280$	3.66487	0.219018
$281$	29.3642	1.75172	0.875860	−	0.482565i	$-0.160295\pi$
$281$	29.3642	1.75172	0.875860	+	0.482565i	$0.160295\pi$
$282$	0	0
$283$	−9.74525	−0.579295	−0.289648	−	0.957133i	$-0.593538\pi$
$283$	−9.74525	−0.579295	−0.289648	+	0.957133i	$0.593538\pi$
$284$	−21.8431	−1.29615
$285$	0	0
$286$	−0.396125	−0.0234233
$287$	−18.1196	−1.06957
$288$	0	0
$289$	−16.9812	−0.998895
$290$	−0.582105	−0.0341824
$291$	0	0
$292$	0.968541	0.0566796
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	9.46011	0.550789
$296$	−0.323044	−0.0187766
$297$	0	0
$298$	44.8853	2.60014
$299$	0.317667	0.0183712
$300$	0	0
$301$	8.15883	0.470267
$302$	0.615957	0.0354443
$303$	0	0
$304$	6.09783	0.349735
$305$	5.10992	0.292593
$306$	0	0
$307$	11.8345	0.675428	0.337714	−	0.941249i	$-0.390346\pi$
$307$	11.8345	0.675428	0.337714	+	0.941249i	$0.390346\pi$
$308$	−4.21983	−0.240447
$309$	0	0
$310$	18.1957	1.03344
$311$	−6.29829	−0.357143	−0.178572	−	0.983927i	$-0.557148\pi$
$311$	−6.29829	−0.357143	−0.178572	+	0.983927i	$0.557148\pi$
$312$	0	0
$313$	31.4620	1.77834	0.889169	−	0.457578i	$-0.151283\pi$
$313$	31.4620	1.77834	0.889169	+	0.457578i	$0.151283\pi$
$314$	−10.9336	−0.617020
$315$	0	0
$316$	38.0030	2.13783
$317$	−3.25773	−0.182973	−0.0914863	−	0.995806i	$-0.529162\pi$
$317$	−3.25773	−0.182973	−0.0914863	+	0.995806i	$0.529162\pi$
$318$	0	0
$319$	0.230586	0.0129103
$320$	13.0368	0.728781
$321$	0	0
$322$	5.60388	0.312292
$323$	1.04221	0.0579903
$324$	0	0
$325$	−0.198062	−0.0109865
$326$	16.4209	0.909468
$327$	0	0
$328$	27.4644	1.51647
$329$	15.0097	0.827511
$330$	0	0
$331$	11.2403	0.617821	0.308911	−	0.951091i	$-0.400036\pi$
$331$	11.2403	0.617821	0.308911	+	0.951091i	$0.400036\pi$
$332$	−29.4142	−1.61431
$333$	0	0
$334$	25.8485	1.41436
$335$	−5.38404	−0.294162
$336$	0	0
$337$	−29.3303	−1.59773	−0.798863	−	0.601513i	$-0.794565\pi$
$337$	−29.3303	−1.59773	−0.798863	+	0.601513i	$0.794565\pi$
$338$	−29.1226	−1.58406
$339$	0	0
$340$	0.417895	0.0226635
$341$	−7.20775	−0.390322
$342$	0	0
$343$	18.0097	0.972432
$344$	−12.3666	−0.666762
$345$	0	0
$346$	5.87263	0.315714
$347$	11.6474	0.625266	0.312633	−	0.949874i	$-0.398789\pi$
$347$	11.6474	0.625266	0.312633	+	0.949874i	$0.398789\pi$
$348$	0	0
$349$	−9.64742	−0.516414	−0.258207	−	0.966090i	$-0.583132\pi$
$349$	−9.64742	−0.516414	−0.258207	+	0.966090i	$0.583132\pi$
$350$	−3.49396	−0.186760
$351$	0	0
$352$	−5.79954	−0.309117
$353$	10.2392	0.544978	0.272489	−	0.962159i	$-0.412153\pi$
$353$	10.2392	0.544978	0.272489	+	0.962159i	$0.412153\pi$
$354$	0	0
$355$	7.16421	0.380237
$356$	3.04892	0.161592
$357$	0	0
$358$	21.8974	1.15731
$359$	−15.0368	−0.793614	−0.396807	−	0.917902i	$-0.629882\pi$
$359$	−15.0368	−0.793614	−0.396807	+	0.917902i	$0.629882\pi$
$360$	0	0
$361$	38.8189	2.04310
$362$	−21.4577	−1.12779
$363$	0	0
$364$	0.939001	0.0492170
$365$	−0.317667	−0.0166275
$366$	0	0
$367$	33.4034	1.74364	0.871822	−	0.489823i	$-0.162938\pi$
$367$	33.4034	1.74364	0.871822	+	0.489823i	$0.162938\pi$
$368$	1.28621	0.0670482
$369$	0	0
$370$	0.307979	0.0160110
$371$	−20.2717	−1.05246
$372$	0	0
$373$	29.6340	1.53439	0.767195	−	0.641414i	$-0.221652\pi$
$373$	29.6340	1.53439	0.767195	+	0.641414i	$0.221652\pi$
$374$	−0.274127	−0.0141748
$375$	0	0
$376$	−22.7506	−1.17327
$377$	−0.0513102	−0.00264261
$378$	0	0
$379$	−14.0629	−0.722364	−0.361182	−	0.932495i	$-0.617627\pi$
$379$	−14.0629	−0.722364	−0.361182	+	0.932495i	$0.617627\pi$
$380$	23.1836	1.18929
$381$	0	0
$382$	−16.9148	−0.865438
$383$	13.1099	0.669885	0.334943	−	0.942238i	$-0.391283\pi$
$383$	13.1099	0.669885	0.334943	+	0.942238i	$0.391283\pi$
$384$	0	0
$385$	1.38404	0.0705374
$386$	11.3177	0.576054
$387$	0	0
$388$	1.34050	0.0680537
$389$	−11.3980	−0.577904	−0.288952	−	0.957344i	$-0.593307\pi$
$389$	−11.3980	−0.577904	−0.288952	+	0.957344i	$0.593307\pi$
$390$	0	0
$391$	0.219833	0.0111174
$392$	−10.7995	−0.545459
$393$	0	0
$394$	0.591794	0.0298141
$395$	−12.4644	−0.627153
$396$	0	0
$397$	−20.5972	−1.03374	−0.516871	−	0.856063i	$-0.672903\pi$
$397$	−20.5972	−1.03374	−0.516871	+	0.856063i	$0.672903\pi$
$398$	−35.7114	−1.79005
$399$	0	0
$400$	−0.801938	−0.0400969
$401$	−20.2150	−1.00949	−0.504746	−	0.863268i	$-0.668414\pi$
$401$	−20.2150	−1.00949	−0.504746	+	0.863268i	$0.668414\pi$
$402$	0	0
$403$	1.60388	0.0798947
$404$	48.1051	2.39332
$405$	0	0
$406$	−0.905149	−0.0449218
$407$	−0.121998	−0.00604721
$408$	0	0
$409$	7.21313	0.356666	0.178333	−	0.983970i	$-0.442930\pi$
$409$	7.21313	0.356666	0.178333	+	0.983970i	$0.442930\pi$
$410$	−26.1836	−1.29312
$411$	0	0
$412$	−50.2814	−2.47719
$413$	14.7101	0.723835
$414$	0	0
$415$	9.64742	0.473573
$416$	1.29052	0.0632730
$417$	0	0
$418$	−15.2078	−0.743835
$419$	2.11290	0.103222	0.0516110	−	0.998667i	$-0.483564\pi$
$419$	2.11290	0.103222	0.0516110	+	0.998667i	$0.483564\pi$
$420$	0	0
$421$	2.82238	0.137554	0.0687772	−	0.997632i	$-0.478090\pi$
$421$	2.82238	0.137554	0.0687772	+	0.997632i	$0.478090\pi$
$422$	−13.2078	−0.642943
$423$	0	0
$424$	30.7265	1.49221
$425$	−0.137063	−0.00664855
$426$	0	0
$427$	7.94571	0.384520
$428$	−18.1739	−0.878469
$429$	0	0
$430$	11.7899	0.568557
$431$	38.8267	1.87022	0.935108	−	0.354363i	$-0.115302\pi$
$431$	38.8267	1.87022	0.935108	+	0.354363i	$0.115302\pi$
$432$	0	0
$433$	−9.66919	−0.464671	−0.232336	−	0.972636i	$-0.574637\pi$
$433$	−9.66919	−0.464671	−0.232336	+	0.972636i	$0.574637\pi$
$434$	28.2935	1.35813
$435$	0	0
$436$	−5.36121	−0.256755
$437$	12.1957	0.583398
$438$	0	0
$439$	26.8310	1.28057	0.640287	−	0.768136i	$-0.278815\pi$
$439$	26.8310	1.28057	0.640287	+	0.768136i	$0.278815\pi$
$440$	−2.09783	−0.100010
$441$	0	0
$442$	0.0609989	0.00290142
$443$	−14.2717	−0.678071	−0.339035	−	0.940774i	$-0.610101\pi$
$443$	−14.2717	−0.678071	−0.339035	+	0.940774i	$0.610101\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	39.9952	1.89383
$447$	0	0
$448$	20.2717	0.957749
$449$	−4.16554	−0.196584	−0.0982920	−	0.995158i	$-0.531338\pi$
$449$	−4.16554	−0.196584	−0.0982920	+	0.995158i	$0.531338\pi$
$450$	0	0
$451$	10.3720	0.488397
$452$	32.9638	1.55048
$453$	0	0
$454$	−14.2620	−0.669351
$455$	−0.307979	−0.0144383
$456$	0	0
$457$	19.7318	0.923017	0.461508	−	0.887136i	$-0.347308\pi$
$457$	19.7318	0.923017	0.461508	+	0.887136i	$0.347308\pi$
$458$	35.7211	1.66914
$459$	0	0
$460$	4.89008	0.228001
$461$	−31.6775	−1.47537	−0.737685	−	0.675145i	$-0.764081\pi$
$461$	−31.6775	−1.47537	−0.737685	+	0.675145i	$0.764081\pi$
$462$	0	0
$463$	−3.31634	−0.154123	−0.0770617	−	0.997026i	$-0.524554\pi$
$463$	−3.31634	−0.154123	−0.0770617	+	0.997026i	$0.524554\pi$
$464$	−0.207751	−0.00964460
$465$	0	0
$466$	26.9584	1.24882
$467$	−6.91185	−0.319842	−0.159921	−	0.987130i	$-0.551124\pi$
$467$	−6.91185	−0.319842	−0.159921	+	0.987130i	$0.551124\pi$
$468$	0	0
$469$	−8.37196	−0.386581
$470$	21.6896	1.00047
$471$	0	0
$472$	−22.2965	−1.02628
$473$	−4.67025	−0.214738
$474$	0	0
$475$	−7.60388	−0.348890
$476$	0.649809	0.0297839
$477$	0	0
$478$	−12.5797	−0.575383
$479$	−18.7633	−0.857317	−0.428659	−	0.903467i	$-0.641014\pi$
$479$	−18.7633	−0.857317	−0.428659	+	0.903467i	$0.641014\pi$
$480$	0	0
$481$	0.0271471	0.00123780
$482$	−26.6896	−1.21568
$483$	0	0
$484$	−31.1226	−1.41466
$485$	−0.439665	−0.0199642
$486$	0	0
$487$	30.3177	1.37382	0.686912	−	0.726740i	$-0.258966\pi$
$487$	30.3177	1.37382	0.686912	+	0.726740i	$0.258966\pi$
$488$	−12.0435	−0.545185
$489$	0	0
$490$	10.2959	0.465121
$491$	−3.75600	−0.169506	−0.0847531	−	0.996402i	$-0.527010\pi$
$491$	−3.75600	−0.169506	−0.0847531	+	0.996402i	$0.527010\pi$
$492$	0	0
$493$	−0.0355078	−0.00159919
$494$	3.38404	0.152255
$495$	0	0
$496$	6.49396	0.291587
$497$	11.1400	0.499699
$498$	0	0
$499$	−29.8431	−1.33596	−0.667980	−	0.744179i	$-0.732841\pi$
$499$	−29.8431	−1.33596	−0.667980	+	0.744179i	$0.732841\pi$
$500$	−3.04892	−0.136352
$501$	0	0
$502$	−36.3913	−1.62422
$503$	−3.26205	−0.145447	−0.0727237	−	0.997352i	$-0.523169\pi$
$503$	−3.26205	−0.145447	−0.0727237	+	0.997352i	$0.523169\pi$
$504$	0	0
$505$	−15.7778	−0.702102
$506$	−3.20775	−0.142602
$507$	0	0
$508$	−54.5217	−2.41901
$509$	−10.9578	−0.485695	−0.242848	−	0.970064i	$-0.578082\pi$
$509$	−10.9578	−0.485695	−0.242848	+	0.970064i	$0.578082\pi$
$510$	0	0
$511$	−0.493959	−0.0218515
$512$	9.00538	0.397985
$513$	0	0
$514$	−3.09783	−0.136640
$515$	16.4916	0.726705
$516$	0	0
$517$	−8.59179	−0.377867
$518$	0.478894	0.0210414
$519$	0	0
$520$	0.466812	0.0204711
$521$	25.5415	1.11900	0.559498	−	0.828832i	$-0.310994\pi$
$521$	25.5415	1.11900	0.559498	+	0.828832i	$0.310994\pi$
$522$	0	0
$523$	−12.7922	−0.559366	−0.279683	−	0.960092i	$-0.590229\pi$
$523$	−12.7922	−0.559366	−0.279683	+	0.960092i	$0.590229\pi$
$524$	16.0435	0.700865
$525$	0	0
$526$	69.6088	3.03509
$527$	1.10992	0.0483487
$528$	0	0
$529$	−20.4276	−0.888156
$530$	−29.2935	−1.27243
$531$	0	0
$532$	36.0495	1.56294
$533$	−2.30798	−0.0999696
$534$	0	0
$535$	5.96077	0.257707
$536$	12.6896	0.548108
$537$	0	0
$538$	−49.5013	−2.13415
$539$	−4.07846	−0.175672
$540$	0	0
$541$	11.6474	0.500762	0.250381	−	0.968147i	$-0.419444\pi$
$541$	11.6474	0.500762	0.250381	+	0.968147i	$0.419444\pi$
$542$	57.1909	2.45656
$543$	0	0
$544$	0.893068	0.0382900
$545$	1.75840	0.0753215
$546$	0	0
$547$	−26.2282	−1.12144	−0.560718	−	0.828007i	$-0.689475\pi$
$547$	−26.2282	−1.12144	−0.560718	+	0.828007i	$0.689475\pi$
$548$	−51.1836	−2.18645
$549$	0	0
$550$	2.00000	0.0852803
$551$	−1.96987	−0.0839192
$552$	0	0
$553$	−19.3817	−0.824192
$554$	27.3056	1.16010
$555$	0	0
$556$	21.0000	0.890598
$557$	−1.60388	−0.0679584	−0.0339792	−	0.999423i	$-0.510818\pi$
$557$	−1.60388	−0.0679584	−0.0339792	+	0.999423i	$0.510818\pi$
$558$	0	0
$559$	1.03923	0.0439547
$560$	−1.24698	−0.0526945
$561$	0	0
$562$	65.9807	2.78323
$563$	3.01208	0.126944	0.0634721	−	0.997984i	$-0.479783\pi$
$563$	3.01208	0.126944	0.0634721	+	0.997984i	$0.479783\pi$
$564$	0	0
$565$	−10.8116	−0.454849
$566$	−21.8974	−0.920416
$567$	0	0
$568$	−16.8853	−0.708491
$569$	−42.5411	−1.78341	−0.891707	−	0.452612i	$-0.850492\pi$
$569$	−42.5411	−1.78341	−0.891707	+	0.452612i	$0.850492\pi$
$570$	0	0
$571$	−5.85862	−0.245176	−0.122588	−	0.992458i	$-0.539119\pi$
$571$	−5.85862	−0.245176	−0.122588	+	0.992458i	$0.539119\pi$
$572$	−0.537500	−0.0224740
$573$	0	0
$574$	−40.7144	−1.69939
$575$	−1.60388	−0.0668862
$576$	0	0
$577$	8.30367	0.345686	0.172843	−	0.984949i	$-0.444705\pi$
$577$	8.30367	0.345686	0.172843	+	0.984949i	$0.444705\pi$
$578$	−38.1564	−1.58710
$579$	0	0
$580$	−0.789856	−0.0327970
$581$	15.0013	0.622360
$582$	0	0
$583$	11.6039	0.480583
$584$	0.748709	0.0309818
$585$	0	0
$586$	−31.4577	−1.29951
$587$	25.9758	1.07214	0.536069	−	0.844174i	$-0.319909\pi$
$587$	25.9758	1.07214	0.536069	+	0.844174i	$0.319909\pi$
$588$	0	0
$589$	61.5749	2.53715
$590$	21.2567	0.875123
$591$	0	0
$592$	0.109916	0.00451753
$593$	−22.3478	−0.917714	−0.458857	−	0.888510i	$-0.651741\pi$
$593$	−22.3478	−0.917714	−0.458857	+	0.888510i	$0.651741\pi$
$594$	0	0
$595$	−0.213128	−0.00873739
$596$	60.9047	2.49475
$597$	0	0
$598$	0.713792	0.0291891
$599$	−8.96316	−0.366225	−0.183112	−	0.983092i	$-0.558617\pi$
$599$	−8.96316	−0.366225	−0.183112	+	0.983092i	$0.558617\pi$
$600$	0	0
$601$	−17.6866	−0.721453	−0.360727	−	0.932672i	$-0.617471\pi$
$601$	−17.6866	−0.721453	−0.360727	+	0.932672i	$0.617471\pi$
$602$	18.3327	0.747186
$603$	0	0
$604$	0.835790	0.0340078
$605$	10.2078	0.415004
$606$	0	0
$607$	−7.45175	−0.302457	−0.151229	−	0.988499i	$-0.548323\pi$
$607$	−7.45175	−0.302457	−0.151229	+	0.988499i	$0.548323\pi$
$608$	49.5448	2.00931
$609$	0	0
$610$	11.4819	0.464888
$611$	1.91185	0.0773453
$612$	0	0
$613$	14.1957	0.573358	0.286679	−	0.958027i	$-0.407449\pi$
$613$	14.1957	0.573358	0.286679	+	0.958027i	$0.407449\pi$
$614$	26.5918	1.07316
$615$	0	0
$616$	−3.26205	−0.131432
$617$	5.98062	0.240771	0.120385	−	0.992727i	$-0.461587\pi$
$617$	5.98062	0.240771	0.120385	+	0.992727i	$0.461587\pi$
$618$	0	0
$619$	−29.5114	−1.18616	−0.593082	−	0.805142i	$-0.702089\pi$
$619$	−29.5114	−1.18616	−0.593082	+	0.805142i	$0.702089\pi$
$620$	24.6896	0.991559
$621$	0	0
$622$	−14.1521	−0.567449
$623$	−1.55496	−0.0622981
$624$	0	0
$625$	1.00000	0.0400000
$626$	70.6945	2.82552
$627$	0	0
$628$	−14.8358	−0.592012
$629$	0.0187864	0.000749061 0
$630$	0	0
$631$	−41.6945	−1.65983	−0.829917	−	0.557888i	$-0.811612\pi$
$631$	−41.6945	−1.65983	−0.829917	+	0.557888i	$0.811612\pi$
$632$	29.3773	1.16857
$633$	0	0
$634$	−7.32006	−0.290717
$635$	17.8823	0.709638
$636$	0	0
$637$	0.907542	0.0359581
$638$	0.518122	0.0205127
$639$	0	0
$640$	16.2620	0.642814
$641$	7.05562	0.278680	0.139340	−	0.990245i	$-0.455502\pi$
$641$	7.05562	0.278680	0.139340	+	0.990245i	$0.455502\pi$
$642$	0	0
$643$	20.6655	0.814966	0.407483	−	0.913213i	$-0.366406\pi$
$643$	20.6655	0.814966	0.407483	+	0.913213i	$0.366406\pi$
$644$	7.60388	0.299635
$645$	0	0
$646$	2.34183	0.0921381
$647$	−22.6353	−0.889887	−0.444943	−	0.895559i	$-0.646776\pi$
$647$	−22.6353	−0.889887	−0.444943	+	0.895559i	$0.646776\pi$
$648$	0	0
$649$	−8.42029	−0.330525
$650$	−0.445042	−0.0174560
$651$	0	0
$652$	22.2814	0.872608
$653$	−23.0315	−0.901291	−0.450645	−	0.892703i	$-0.648806\pi$
$653$	−23.0315	−0.901291	−0.450645	+	0.892703i	$0.648806\pi$
$654$	0	0
$655$	−5.26205	−0.205605
$656$	−9.34481	−0.364854
$657$	0	0
$658$	33.7265	1.31479
$659$	−34.0388	−1.32596	−0.662981	−	0.748636i	$-0.730709\pi$
$659$	−34.0388	−1.32596	−0.662981	+	0.748636i	$0.730709\pi$
$660$	0	0
$661$	−23.3405	−0.907840	−0.453920	−	0.891042i	$-0.649975\pi$
$661$	−23.3405	−0.907840	−0.453920	+	0.891042i	$0.649975\pi$
$662$	25.2567	0.981628
$663$	0	0
$664$	−22.7380	−0.882404
$665$	−11.8237	−0.458504
$666$	0	0
$667$	−0.415502	−0.0160883
$668$	35.0737	1.35704
$669$	0	0
$670$	−12.0978	−0.467380
$671$	−4.54825	−0.175583
$672$	0	0
$673$	−41.4905	−1.59934	−0.799671	−	0.600439i	$-0.794993\pi$
$673$	−41.4905	−1.59934	−0.799671	+	0.600439i	$0.794993\pi$
$674$	−65.9047	−2.53855
$675$	0	0
$676$	−39.5163	−1.51986
$677$	−49.7706	−1.91284	−0.956420	−	0.291995i	$-0.905681\pi$
$677$	−49.7706	−1.91284	−0.956420	+	0.291995i	$0.905681\pi$
$678$	0	0
$679$	−0.683661	−0.0262365
$680$	0.323044	0.0123882
$681$	0	0
$682$	−16.1957	−0.620164
$683$	42.9965	1.64522	0.822608	−	0.568608i	$-0.192518\pi$
$683$	42.9965	1.64522	0.822608	+	0.568608i	$0.192518\pi$
$684$	0	0
$685$	16.7875	0.641416
$686$	40.4674	1.54505
$687$	0	0
$688$	4.20775	0.160419
$689$	−2.58211	−0.0983704
$690$	0	0
$691$	20.1196	0.765386	0.382693	−	0.923876i	$-0.374997\pi$
$691$	20.1196	0.765386	0.382693	+	0.923876i	$0.374997\pi$
$692$	7.96854	0.302918
$693$	0	0
$694$	26.1715	0.993457
$695$	−6.88769	−0.261265
$696$	0	0
$697$	−1.59717	−0.0604972
$698$	−21.6775	−0.820507
$699$	0	0
$700$	−4.74094	−0.179191
$701$	−7.23191	−0.273146	−0.136573	−	0.990630i	$-0.543609\pi$
$701$	−7.23191	−0.273146	−0.136573	+	0.990630i	$0.543609\pi$
$702$	0	0
$703$	1.04221	0.0393078
$704$	−11.6039	−0.437338
$705$	0	0
$706$	23.0073	0.865891
$707$	−24.5338	−0.922688
$708$	0	0
$709$	42.8370	1.60878	0.804388	−	0.594104i	$-0.202493\pi$
$709$	42.8370	1.60878	0.804388	+	0.594104i	$0.202493\pi$
$710$	16.0978	0.604141
$711$	0	0
$712$	2.35690	0.0883284
$713$	12.9879	0.486401
$714$	0	0
$715$	0.176292	0.00659295
$716$	29.7125	1.11041
$717$	0	0
$718$	−33.7875	−1.26094
$719$	−50.9922	−1.90169	−0.950845	−	0.309668i	$-0.899782\pi$
$719$	−50.9922	−1.90169	−0.950845	+	0.309668i	$0.899782\pi$
$720$	0	0
$721$	25.6437	0.955021
$722$	87.2253	3.24619
$723$	0	0
$724$	−29.1159	−1.08208
$725$	0.259061	0.00962129
$726$	0	0
$727$	−20.2892	−0.752484	−0.376242	−	0.926521i	$-0.622784\pi$
$727$	−20.2892	−0.752484	−0.376242	+	0.926521i	$0.622784\pi$
$728$	0.725873	0.0269027
$729$	0	0
$730$	−0.713792	−0.0264186
$731$	0.719169	0.0265994
$732$	0	0
$733$	−8.90946	−0.329078	−0.164539	−	0.986371i	$-0.552614\pi$
$733$	−8.90946	−0.329078	−0.164539	+	0.986371i	$0.552614\pi$
$734$	75.0568	2.77040
$735$	0	0
$736$	10.4504	0.385208
$737$	4.79225	0.176525
$738$	0	0
$739$	2.60255	0.0957363	0.0478681	−	0.998854i	$-0.484757\pi$
$739$	2.60255	0.0957363	0.0478681	+	0.998854i	$0.484757\pi$
$740$	0.417895	0.0153621
$741$	0	0
$742$	−45.5502	−1.67220
$743$	41.3793	1.51806	0.759029	−	0.651057i	$-0.225674\pi$
$743$	41.3793	1.51806	0.759029	+	0.651057i	$0.225674\pi$
$744$	0	0
$745$	−19.9758	−0.731858
$746$	66.5870	2.43792
$747$	0	0
$748$	−0.371961	−0.0136003
$749$	9.26875	0.338673
$750$	0	0
$751$	12.2457	0.446850	0.223425	−	0.974721i	$-0.428276\pi$
$751$	12.2457	0.446850	0.223425	+	0.974721i	$0.428276\pi$
$752$	7.74094	0.282283
$753$	0	0
$754$	−0.115293	−0.00419873
$755$	−0.274127	−0.00997649
$756$	0	0
$757$	−45.7646	−1.66334	−0.831672	−	0.555267i	$-0.812616\pi$
$757$	−45.7646	−1.66334	−0.831672	+	0.555267i	$0.812616\pi$
$758$	−31.5991	−1.14773
$759$	0	0
$760$	17.9215	0.650083
$761$	16.6944	0.605172	0.302586	−	0.953122i	$-0.402150\pi$
$761$	16.6944	0.605172	0.302586	+	0.953122i	$0.402150\pi$
$762$	0	0
$763$	2.73423	0.0989859
$764$	−22.9517	−0.830362
$765$	0	0
$766$	29.4577	1.06435
$767$	1.87369	0.0676550
$768$	0	0
$769$	−3.49694	−0.126103	−0.0630515	−	0.998010i	$-0.520083\pi$
$769$	−3.49694	−0.126103	−0.0630515	+	0.998010i	$0.520083\pi$
$770$	3.10992	0.112074
$771$	0	0
$772$	15.3569	0.552707
$773$	−24.5810	−0.884119	−0.442059	−	0.896986i	$-0.645752\pi$
$773$	−24.5810	−0.884119	−0.442059	+	0.896986i	$0.645752\pi$
$774$	0	0
$775$	−8.09783	−0.290883
$776$	1.03624	0.0371990
$777$	0	0
$778$	−25.6112	−0.918205
$779$	−88.6064	−3.17465
$780$	0	0
$781$	−6.37675	−0.228178
$782$	0.493959	0.0176639
$783$	0	0
$784$	3.67456	0.131234
$785$	4.86592	0.173672
$786$	0	0
$787$	−12.5670	−0.447967	−0.223983	−	0.974593i	$-0.571906\pi$
$787$	−12.5670	−0.447967	−0.223983	+	0.974593i	$0.571906\pi$
$788$	0.803003	0.0286058
$789$	0	0
$790$	−28.0073	−0.996455
$791$	−16.8116	−0.597753
$792$	0	0
$793$	1.01208	0.0359401
$794$	−46.2814	−1.64247
$795$	0	0
$796$	−48.4566	−1.71750
$797$	9.38644	0.332485	0.166242	−	0.986085i	$-0.446837\pi$
$797$	9.38644	0.332485	0.166242	+	0.986085i	$0.446837\pi$
$798$	0	0
$799$	1.32304	0.0468059
$800$	−6.51573	−0.230366
$801$	0	0
$802$	−45.4228	−1.60393
$803$	0.282750	0.00997805
$804$	0	0
$805$	−2.49396	−0.0879005
$806$	3.60388	0.126941
$807$	0	0
$808$	37.1866	1.30822
$809$	12.6461	0.444613	0.222306	−	0.974977i	$-0.428641\pi$
$809$	12.6461	0.444613	0.222306	+	0.974977i	$0.428641\pi$
$810$	0	0
$811$	−3.33619	−0.117149	−0.0585747	−	0.998283i	$-0.518656\pi$
$811$	−3.33619	−0.117149	−0.0585747	+	0.998283i	$0.518656\pi$
$812$	−1.22819	−0.0431011
$813$	0	0
$814$	−0.274127	−0.00960814
$815$	−7.30798	−0.255987
$816$	0	0
$817$	39.8974	1.39583
$818$	16.2078	0.566691
$819$	0	0
$820$	−35.5284	−1.24071
$821$	46.2258	1.61329	0.806646	−	0.591035i	$-0.201281\pi$
$821$	46.2258	1.61329	0.806646	+	0.591035i	$0.201281\pi$
$822$	0	0
$823$	−23.1836	−0.808129	−0.404065	−	0.914730i	$-0.632403\pi$
$823$	−23.1836	−0.808129	−0.404065	+	0.914730i	$0.632403\pi$
$824$	−38.8689	−1.35406
$825$	0	0
$826$	33.0532	1.15007
$827$	−46.8224	−1.62817	−0.814087	−	0.580743i	$-0.802762\pi$
$827$	−46.8224	−1.62817	−0.814087	+	0.580743i	$0.802762\pi$
$828$	0	0
$829$	10.6896	0.371266	0.185633	−	0.982619i	$-0.440566\pi$
$829$	10.6896	0.371266	0.185633	+	0.982619i	$0.440566\pi$
$830$	21.6775	0.752439
$831$	0	0
$832$	2.58211	0.0895184
$833$	0.628039	0.0217602
$834$	0	0
$835$	−11.5036	−0.398100
$836$	−20.6353	−0.713688
$837$	0	0
$838$	4.74764	0.164005
$839$	−26.6079	−0.918608	−0.459304	−	0.888279i	$-0.651901\pi$
$839$	−26.6079	−0.918608	−0.459304	+	0.888279i	$0.651901\pi$
$840$	0	0
$841$	−28.9329	−0.997686
$842$	6.34183	0.218554
$843$	0	0
$844$	−17.9215	−0.616885
$845$	12.9608	0.445864
$846$	0	0
$847$	15.8726	0.545390
$848$	−10.4547	−0.359017
$849$	0	0
$850$	−0.307979	−0.0105636
$851$	0.219833	0.00753576
$852$	0	0
$853$	−10.2698	−0.351632	−0.175816	−	0.984423i	$-0.556256\pi$
$853$	−10.2698	−0.351632	−0.175816	+	0.984423i	$0.556256\pi$
$854$	17.8538	0.610946
$855$	0	0
$856$	−14.0489	−0.480182
$857$	13.0965	0.447368	0.223684	−	0.974662i	$-0.428192\pi$
$857$	13.0965	0.447368	0.223684	+	0.974662i	$0.428192\pi$
$858$	0	0
$859$	−13.5711	−0.463040	−0.231520	−	0.972830i	$-0.574370\pi$
$859$	−13.5711	−0.463040	−0.231520	+	0.972830i	$0.574370\pi$
$860$	15.9976	0.545514
$861$	0	0
$862$	87.2428	2.97150
$863$	−0.434879	−0.0148035	−0.00740173	−	0.999973i	$-0.502356\pi$
$863$	−0.434879	−0.0148035	−0.00740173	+	0.999973i	$0.502356\pi$
$864$	0	0
$865$	−2.61356	−0.0888638
$866$	−21.7265	−0.738295
$867$	0	0
$868$	38.3913	1.30309
$869$	11.0944	0.376351
$870$	0	0
$871$	−1.06638	−0.0361328
$872$	−4.14436	−0.140346
$873$	0	0
$874$	27.4034	0.926935
$875$	1.55496	0.0525672
$876$	0	0
$877$	5.09916	0.172186	0.0860932	−	0.996287i	$-0.472562\pi$
$877$	5.09916	0.172186	0.0860932	+	0.996287i	$0.472562\pi$
$878$	60.2887	2.03465
$879$	0	0
$880$	0.713792	0.0240619
$881$	−41.6582	−1.40350	−0.701750	−	0.712424i	$-0.747598\pi$
$881$	−41.6582	−1.40350	−0.701750	+	0.712424i	$0.747598\pi$
$882$	0	0
$883$	−22.8955	−0.770494	−0.385247	−	0.922814i	$-0.625884\pi$
$883$	−22.8955	−0.770494	−0.385247	+	0.922814i	$0.625884\pi$
$884$	0.0827692	0.00278383
$885$	0	0
$886$	−32.0683	−1.07736
$887$	−4.93362	−0.165655	−0.0828274	−	0.996564i	$-0.526395\pi$
$887$	−4.93362	−0.165655	−0.0828274	+	0.996564i	$0.526395\pi$
$888$	0	0
$889$	27.8062	0.932592
$890$	−2.24698	−0.0753189
$891$	0	0
$892$	54.2693	1.81707
$893$	73.3986	2.45619
$894$	0	0
$895$	−9.74525	−0.325748
$896$	25.2868	0.844773
$897$	0	0
$898$	−9.35988	−0.312343
$899$	−2.09783	−0.0699667
$900$	0	0
$901$	−1.78687	−0.0595293
$902$	23.3056	0.775991
$903$	0	0
$904$	25.4819	0.847515
$905$	9.54958	0.317439
$906$	0	0
$907$	17.8043	0.591183	0.295592	−	0.955314i	$-0.404483\pi$
$907$	17.8043	0.591183	0.295592	+	0.955314i	$0.404483\pi$
$908$	−19.3521	−0.642222
$909$	0	0
$910$	−0.692021	−0.0229403
$911$	−12.9879	−0.430309	−0.215154	−	0.976580i	$-0.569025\pi$
$911$	−12.9879	−0.430309	−0.215154	+	0.976580i	$0.569025\pi$
$912$	0	0
$913$	−8.58701	−0.284188
$914$	44.3370	1.46654
$915$	0	0
$916$	48.4698	1.60149
$917$	−8.18226	−0.270202
$918$	0	0
$919$	24.3961	0.804754	0.402377	−	0.915474i	$-0.368184\pi$
$919$	24.3961	0.804754	0.402377	+	0.915474i	$0.368184\pi$
$920$	3.78017	0.124628
$921$	0	0
$922$	−71.1788	−2.34415
$923$	1.41896	0.0467056
$924$	0	0
$925$	−0.137063	−0.00450661
$926$	−7.45175	−0.244880
$927$	0	0
$928$	−1.68797	−0.0554104
$929$	−27.1400	−0.890436	−0.445218	−	0.895422i	$-0.646874\pi$
$929$	−27.1400	−0.890436	−0.445218	+	0.895422i	$0.646874\pi$
$930$	0	0
$931$	34.8418	1.14189
$932$	36.5797	1.19821
$933$	0	0
$934$	−15.5308	−0.508183
$935$	0.121998	0.00398976
$936$	0	0
$937$	44.9241	1.46760	0.733802	−	0.679363i	$-0.237744\pi$
$937$	44.9241	1.46760	0.733802	+	0.679363i	$0.237744\pi$
$938$	−18.8116	−0.614221
$939$	0	0
$940$	29.4306	0.959919
$941$	6.40044	0.208648	0.104324	−	0.994543i	$-0.466732\pi$
$941$	6.40044	0.208648	0.104324	+	0.994543i	$0.466732\pi$
$942$	0	0
$943$	−18.6896	−0.608618
$944$	7.58642	0.246917
$945$	0	0
$946$	−10.4940	−0.341188
$947$	14.2067	0.461655	0.230828	−	0.972995i	$-0.425857\pi$
$947$	14.2067	0.461655	0.230828	+	0.972995i	$0.425857\pi$
$948$	0	0
$949$	−0.0629179	−0.00204240
$950$	−17.0858	−0.554335
$951$	0	0
$952$	0.502320	0.0162803
$953$	−32.2801	−1.04565	−0.522827	−	0.852439i	$-0.675123\pi$
$953$	−32.2801	−1.04565	−0.522827	+	0.852439i	$0.675123\pi$
$954$	0	0
$955$	7.52781	0.243594
$956$	−17.0694	−0.552063
$957$	0	0
$958$	−42.1608	−1.36215
$959$	26.1038	0.842936
$960$	0	0
$961$	34.5749	1.11532
$962$	0.0609989	0.00196668
$963$	0	0
$964$	−36.2150	−1.16641
$965$	−5.03684	−0.162141
$966$	0	0
$967$	30.2067	0.971382	0.485691	−	0.874131i	$-0.338568\pi$
$967$	30.2067	0.971382	0.485691	+	0.874131i	$0.338568\pi$
$968$	−24.0586	−0.773273
$969$	0	0
$970$	−0.987918	−0.0317201
$971$	43.2137	1.38679	0.693397	−	0.720556i	$-0.256113\pi$
$971$	43.2137	1.38679	0.693397	+	0.720556i	$0.256113\pi$
$972$	0	0
$973$	−10.7101	−0.343349
$974$	68.1232	2.18281
$975$	0	0
$976$	4.09783	0.131168
$977$	51.5997	1.65082	0.825410	−	0.564534i	$-0.190944\pi$
$977$	51.5997	1.65082	0.825410	+	0.564534i	$0.190944\pi$
$978$	0	0
$979$	0.890084	0.0284472
$980$	13.9705	0.446270
$981$	0	0
$982$	−8.43967	−0.269321
$983$	−52.9934	−1.69023	−0.845114	−	0.534587i	$-0.820467\pi$
$983$	−52.9934	−1.69023	−0.845114	+	0.534587i	$0.820467\pi$
$984$	0	0
$985$	−0.263373	−0.00839176
$986$	−0.0797853	−0.00254088
$987$	0	0
$988$	4.59179	0.146084
$989$	8.41550	0.267597
$990$	0	0
$991$	2.48321	0.0788816	0.0394408	−	0.999222i	$-0.487442\pi$
$991$	2.48321	0.0788816	0.0394408	+	0.999222i	$0.487442\pi$
$992$	52.7633	1.67524
$993$	0	0
$994$	25.0315	0.793950
$995$	15.8931	0.503844
$996$	0	0
$997$	−3.62325	−0.114750	−0.0573748	−	0.998353i	$-0.518273\pi$
$997$	−3.62325	−0.114750	−0.0573748	+	0.998353i	$0.518273\pi$
$998$	−67.0568	−2.12265
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.j.1.3		3
3.2	odd	2		1335.2.a.d.1.1	✓	3
15.14	odd	2		6675.2.a.o.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.d.1.1	✓	3	3.2	odd	2
4005.2.a.j.1.3		3	1.1	even	1	trivial
6675.2.a.o.1.3		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+2.24698 q^{2} +3.04892 q^{4} -1.00000 q^{5} -1.55496 q^{7} +2.35690 q^{8} +O(q^{10})\)	\(q+2.24698 q^{2} +3.04892 q^{4} -1.00000 q^{5} -1.55496 q^{7} +2.35690 q^{8} -2.24698 q^{10} +0.890084 q^{11} -0.198062 q^{13} -3.49396 q^{14} -0.801938 q^{16} -0.137063 q^{17} -7.60388 q^{19} -3.04892 q^{20} +2.00000 q^{22} -1.60388 q^{23} +1.00000 q^{25} -0.445042 q^{26} -4.74094 q^{28} +0.259061 q^{29} -8.09783 q^{31} -6.51573 q^{32} -0.307979 q^{34} +1.55496 q^{35} -0.137063 q^{37} -17.0858 q^{38} -2.35690 q^{40} +11.6528 q^{41} -5.24698 q^{43} +2.71379 q^{44} -3.60388 q^{46} -9.65279 q^{47} -4.58211 q^{49} +2.24698 q^{50} -0.603875 q^{52} +13.0368 q^{53} -0.890084 q^{55} -3.66487 q^{56} +0.582105 q^{58} -9.46011 q^{59} -5.10992 q^{61} -18.1957 q^{62} -13.0368 q^{64} +0.198062 q^{65} +5.38404 q^{67} -0.417895 q^{68} +3.49396 q^{70} -7.16421 q^{71} +0.317667 q^{73} -0.307979 q^{74} -23.1836 q^{76} -1.38404 q^{77} +12.4644 q^{79} +0.801938 q^{80} +26.1836 q^{82} -9.64742 q^{83} +0.137063 q^{85} -11.7899 q^{86} +2.09783 q^{88} +1.00000 q^{89} +0.307979 q^{91} -4.89008 q^{92} -21.6896 q^{94} +7.60388 q^{95} +0.439665 q^{97} -10.2959 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} - 3 q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 - 3 * q^5 - 5 * q^7 + 3 * q^8	\( 3 q + 2 q^{2} - 3 q^{5} - 5 q^{7} + 3 q^{8} - 2 q^{10} + 2 q^{11} - 5 q^{13} - q^{14} + 2 q^{16} + 5 q^{17} - 14 q^{19} + 6 q^{22} + 4 q^{23} + 3 q^{25} - q^{26} + 15 q^{29} - 6 q^{31} - 7 q^{32} - 6 q^{34} + 5 q^{35} + 5 q^{37} - 14 q^{38} - 3 q^{40} + 17 q^{41} - 11 q^{43} - 2 q^{46} - 11 q^{47} - 8 q^{49} + 2 q^{50} + 7 q^{52} + 11 q^{53} - 2 q^{55} - 12 q^{56} - 4 q^{58} - 3 q^{59} - 16 q^{61} - 18 q^{62} - 11 q^{64} + 5 q^{65} + 6 q^{67} - 7 q^{68} + q^{70} - 10 q^{71} - 16 q^{73} - 6 q^{74} - 14 q^{76} + 6 q^{77} - 7 q^{79} - 2 q^{80} + 23 q^{82} - 14 q^{83} - 5 q^{85} - 12 q^{86} - 12 q^{88} + 3 q^{89} + 6 q^{91} - 14 q^{92} - 19 q^{94} + 14 q^{95} + 4 q^{97} - 17 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 - 3 * q^5 - 5 * q^7 + 3 * q^8 - 2 * q^10 + 2 * q^11 - 5 * q^13 - q^14 + 2 * q^16 + 5 * q^17 - 14 * q^19 + 6 * q^22 + 4 * q^23 + 3 * q^25 - q^26 + 15 * q^29 - 6 * q^31 - 7 * q^32 - 6 * q^34 + 5 * q^35 + 5 * q^37 - 14 * q^38 - 3 * q^40 + 17 * q^41 - 11 * q^43 - 2 * q^46 - 11 * q^47 - 8 * q^49 + 2 * q^50 + 7 * q^52 + 11 * q^53 - 2 * q^55 - 12 * q^56 - 4 * q^58 - 3 * q^59 - 16 * q^61 - 18 * q^62 - 11 * q^64 + 5 * q^65 + 6 * q^67 - 7 * q^68 + q^70 - 10 * q^71 - 16 * q^73 - 6 * q^74 - 14 * q^76 + 6 * q^77 - 7 * q^79 - 2 * q^80 + 23 * q^82 - 14 * q^83 - 5 * q^85 - 12 * q^86 - 12 * q^88 + 3 * q^89 + 6 * q^91 - 14 * q^92 - 19 * q^94 + 14 * q^95 + 4 * q^97 - 17 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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